Let $K$ be a field of positive characteristic $p$. Consider the system of $\mathbb F_p$-linear equations $$\left\{\begin{array}{ccl} a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&b_1\\ a_{11}x^p_1+a_{12}x^p_2+\cdots+a_{1n}x^p_n&=&b_2\\ \vdots&\vdots&\vdots\\ a_{n1}x^{p^{n-1}}_1+a_{n2}x^{p^{n-1}}_2+\cdots+a_{nn}x^{p^{n-1}}_n&=&b_n\end{array}\right.$$ where tha $a_{i,j}$'s and $b_i$'s are in $K$ and the unknowns are $x_1,\cdots,x_n$. Is there a tool (as the determinant in the usual case) that characterizes the fact that this system has a unique solution $(x_1,\cdots,x_n)$ in an algebraic closure of $K$. I know Moore determinant but I do not see how to apply it to my problem. I know Grobner bases, but in the theorical case like this one, it looks like it is pretty useless.
$\begingroup$
$\endgroup$
3
-
6$\begingroup$ In an algebraic closure $L$ of $K$, can't you just write $a_{ij}=c_{ij}^{p^i}$ and $b_i=d_i^{p^i}$ for (unique) $c_{ij}$ and $d_i$, and then you're $i$th equation becomes $$(c_{i1}x_1+\cdots+c_{in}x_n)^{p^i}=d_i^{p^i},$$ which in turn is equivalent to $$c_{i1}x_1+\cdots+c_{in}x^n=d_i.$$ So the existence of a unique solution is $\det(c_{ij})\ne0$. (Not sure if this helps, though.) $\endgroup$– Joe SilvermanSep 6, 2020 at 1:46
-
$\begingroup$ Please correct the subscripts in the second equation. $\endgroup$– Wilberd van der KallenSep 6, 2020 at 6:56
-
1$\begingroup$ @JoeSilverman we could also consider $p^{n-k}$-th power of the $k$-th equation, it would give a linear system on $x_1^p, ... x_n^p$. Then, det of this system would guarantee existence, and we would need to find $p^{n-1}$-th root to solve the original system. It also gives the criterion when the solution is in the base field I think? $\endgroup$– Lev SoukhanovSep 6, 2020 at 11:32
Add a comment
|